Adaptive beamforming is a powerful technique used in modern radars to mitigate the impact of unintentional interference and hostile jamming. Typically, nulls are created in the receive pattern of an array by applying a complex weight to each array element. Using conventional linear processing, an array of N physical elements can form no more than N−1 adaptive nulls. To overcome this limitation, nonlinear techniques have been developed capable of forming O(N2) nulls in an array pattern. A drawback to nonlinear adaptive processing is that any Doppler information in the received signal is lost.
To provide a brief overview of nonlinear adaptive processing consider an N element nonuniform linear array (NULA). Assume M narrowband signals arc arriving at this array from directions 01, 02, . . . , 0M with powers σ12, σ22, . . . , σM2, respectively. Let v(0) be the N-by-1 steering vector corresponding to the direction 0,v(0)=[1ej(2π/λ)dt sin0 . . . ej(2π/λ)dN1 sin0 ]T   (1)where d1 denotes the position of the ith sensor. The received signal at time instant k isx[k]=A(0)s(k)+n[k]  (2)where A(0)=[v(01) v(02) . . . v(0M)] is the array manifold matrix and s[k]=[s1[k] s2[k] . . . sm [k]]T is a vector of samples from uncorrelated signal sources. The noise n[k] is assumed to be temporally uncorrelated so that the signal covariance matrix Rss is diagonal. Now the covariance matrix of the received signal becomes
                                                                        R                zz                            =                                                E                  ⁡                                      [                                          xx                      H                                        ]                                                  =                                                                            A                      ⁡                                              (                        0                        )                                                              ⁢                                          R                      ss                                        ⁢                                                                  A                        ⁡                                                  (                          0                          )                                                                    H                                                        +                                                            σ                      n                      2                                        ⁢                    I                                                                                                                          =                                                                                          A                      ⁡                                              (                        0                        )                                                              ⁡                                          [                                                                                                                                  σ                              1                              2                                                                                                            0                                                                                …                                                                                0                                                                                                                                0                                                                                                              σ                              2                              2                                                                                                            ⋱                                                                                ⋮                                                                                                                                ⋮                                                                                ⋱                                                                                ⋱                                                                                0                                                                                                                                0                                                                                …                                                                                0                                                                                                              σ                              M                              2                                                                                                                          ]                                                        ⁢                                                            A                      ⁡                                              (                        0                        )                                                              H                                                  +                                                      σ                    n                    2                                    ⁢                                      I                    .                                                                                                                                              (              3              )                                                                          (              4              )                                          Next, the covariance matrix Rss is vectorited to create the vector
                                                                        z                ⁡                                  (                  0                  )                                            =                                                vcc                  ⁡                                      (                                          R                      zz                                        )                                                  =                                                      vcc                    ⁡                                          [                                                                        ∑                                                      i                            =                            1                                                    M                                                ⁢                                                                              σ                            i                            2                                                    ⁡                                                      (                                                                                          v                                ⁡                                                                  (                                                                      θ                                    i                                                                    )                                                                                            ⁢                                                                                                v                                  ⁡                                                                      (                                                                          θ                                      i                                                                        )                                                                                                  H                                                                                      )                                                                                              ]                                                        +                                                            σ                      n                      2                                        ⁢                                          1                      n                                                                                                                                              =                                                                    (                                                                                            A                          ⁡                                                      (                            0                            )                                                                          *                                            ⊙                                              A                        ⁡                                                  (                          0                          )                                                                                      )                                    ⁢                  p                                +                                                      σ                    n                    2                                    ⁢                                      1                    n                                                                                                                                              (              5              )                                                                                                                                                                    (                6                )                            ⁢                                                                                      where * denotes conjugation. p=[σ12 σ22 . . . σM2]T and 1n=[e1T e2T . . . eNT]T with ei a column vector of all zeros except for a one in the ith position. The matrixA(0)*: A(0)=[v(01)*{circle around (⋅)}v(01) . . . v(0M)*{circle around (⋅)}v(0M)]  (7)is the Khatri-Rao product of the matrices A(0)* and A(0) with {circle around (⋅)} denoting the Kronecker product. In conventional nonlinear adaptive processing. the adapted beampattern is formed by applying a weight vector w to the vector z(0): as in wHz(0). [1]-[3].
A drawback to nonlinear adaptive processing is that any Doppler information in the received signal is lost.